Congruence II Proposition List

    This topic proves the well-known Alternate Interior Angle and Exterior Angle Propositions much as Euclid did for the Euclidean plane. However, Euclid had gaps in these proofs; in fact, his gaps may not be noticed until trying to follow his proofs on a sphere. This topic uses the Crossbar Proposition to close the gap in the Exterior Angle Proposition. Also the SAA Congruence Criterion and the Hypotenuse-Leg Congruence Criterion are detailed. Propositions concerning midpoints, bisectors, and relations between sides and angles of a triangle are proven.

Proposition (Alternate Interior Angle)
    (i) If two lines cut by a transversal have a pair of congruent alternate interior angles, then the two lines are parrallel.
    (ii) Two lines perpendicular to the same line are parallel.
    (iii) If congruence ii proposition list _gr_1.gif] is any line and congruence ii proposition list _gr_2.gif] is any point not on congruence ii proposition list _gr_3.gif] there exists at least one line congruence ii proposition list _gr_4.gif] through congruence ii proposition list _gr_5.gif] parallel to congruence ii proposition list _gr_6.gif]

Proposition (Exterior Angle Property) An exterior angle of a triangle is greater than either remote interior angle.

Proposition (SAA Congruence Criterion) Given congruence ii proposition list _gr_7.gif] congruence ii proposition list _gr_8.gif] and congruence ii proposition list _gr_9.gif] Then congruence ii proposition list _gr_10.gif]

Proposition (Hypotenuse-Leg Criterion) Two right triangles are congruent if the hypotneuse and a leg of one are congruent repsectively to the hypotenuse and a leg of the other.

Proposition (Midpoints) Every segment has a unique midpoint.

Proposition (Bisectors)
    (i) Every angle has a unqiue bisector
    (ii) Every segment has a unique perpendicular bisector.

Proposition (Larger Angle Larger Side) In a triangle congruence ii proposition list _gr_11.gif] the greater angle lies opposite the greater side and the greater side lies opposite the greater angle.

Proposition (Side Angle Comparison) Given congruence ii proposition list _gr_12.gif] and   congruence ii proposition list _gr_13.gif] if congruence ii proposition list _gr_14.gif] and congruence ii proposition list _gr_15.gif] then congruence ii proposition list _gr_16.gif] if and only if congruence ii proposition list _gr_17.gif]

Proposition (Segment Comparison)
    (i) If congruence ii proposition list _gr_18.gif] and congruence ii proposition list _gr_19.gif] then congruence ii proposition list _gr_20.gif]
    (ii) Given any triangle congruence ii proposition list _gr_21.gif] and point congruence ii proposition list _gr_22.gif] such that congruence ii proposition list _gr_23.gif] then congruence ii proposition list _gr_24.gif] or congruence ii proposition list _gr_25.gif]

Cite this as:
Congruence Ii Proposition List
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/congruence-ii-proposition-list.html
 
    
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